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where σsf denotes the surface charge density and jel,sf the surface current, as yet undetermined. Combining Eqs(78) and (84), we obtain 0=∆(Qn + Q D n +(µ + c 2 ) j D n − (µ + c 2 ) ρvn) . (94) which in conjunction with Eqs(49, 15, 45) implies 0=∆ (T s+ v · g) vn − Π D nj vj − Tf D n +c E × H + E D × H 0 + E 0 × H D · n . (95) If the interface is curved, we shall from here on choose the inertial system vt = 0 (which gets rid of a term ∼ vtαsf in R sf ), otherwise, vt is arbitrary. As vn is negligibly small in the absence of phase transition, Eq(95) is quickly converted into the surface entropy production, R sf ≡−T ∆fn = fn ∆T − Π D jn ∆vt,j +c ∆ E × H + E D × H 0 + E 0 × H D · n (96) with fn ≡ (svn − f D n ), here and below. The surface entropy production R sf ≡ −T ∆fn is the difference between the entropy current exiting and entering the surface. Just as its bulk counter part, it is positive definite, invariant under time inversion and vanishes in equilibrium. Some more algebra then yields R sf = fn ∆T + c n × E D · ∆H + c H D × n · ∆E +(−Π D jn + H D j Bn + E D j Dn + 1 4 σsf∆Ej)∆vt,j +[n × jel,sf] · (n × E 0 ) , (97) a nice sum of independent pairs of force and fluxes. Taking them to be proportional to each other, they represent another seven boundary conditions, and determine the surface current jel,sf. One symmetry element peculiar to the interface is the invariance of R sf under the simultaneous operation n → −n and ∆ → −∆. Taking this and the isotropy into account, we find ⎛ fn = κs∆T ⎞ ⎛ (98) ⎜ n × ED ⎟ ⎜ ⎝ ⎠ = ⎝ n × jel,sf βs ⎞ ⎛ γs ⎟ ⎜ c ∆Ht ⎠ ⎝ ¯γs σs n × E0 ⎞ ⎟ ⎠ , (99) 16
⎛ ⎜ ⎝ −ΠD,eff t,i HD × n ⎞ ⎛ ⎟ ⎜ ⎠ = ⎝ i ηs ⎞ ⎛ ζs ⎟ ⎜ ⎠ ⎝ ¯ζs αs ∆vt,i ⎞ ⎟ ⎠ c ∆Et,i (100) where −Π D,eff t,i ≡−Π D t,i + H D t,i Bn + E D t,i Dn + 1 4 σsf∆Et,j, and ¯γs = −γs , ¯ ζs = −ζs . (101) αs,βs,ηs,κs,σs > 0 . (102) These are altogether 21 boundary conditions. They suffice to determine (i) all (outgoing) hydrodynamic modes of the dielectric medium, 9 for each side; (ii) the normal components of B + B D and D + D D , (iii) the lab velocity. See [14] for a more detailed consideration (in single-component liquids). 4.2 The Dielectric-Dielectric Interface, with Phase Transition Mass and concentration currents across the interface renders the consideration slightly more complicated. The 12 continuity conditions remain: ∆(Qn + Q D n )=0, (103) ∆(ρvn − j D n )=0, (104) ∆(ρc vn − j D c,n) = 0 (105) ∆(Πnn − Π D nn) =Psf , (106) ∆(Πt,i − Π D t,i) t1,i = t1 ·∇αsf , (107) ∆(Πt,i − Π D t,i) t2,i = t2 ·∇αsf (108) ∆(B D n + Bn) =0, (109) ∆(Et + E D t )=0, (110) ∆(D D n + Dn) =−σsf , (111) ∆(Ht + H D t )=n × jel,sf/c . (112) The rest of 9 boundary conditions must now be deduced from R sf . Combining Eq(103) and (104), we obtain 0=∆(Qn + Q D n ) − (µ + c 2 )∆(ρvn − j D n ) =∆(Qn + Q D n +(µ + c 2 ) j D n − (µ + c 2 ) ρvn) +〈ρvn − j D n 〉∆µ, (113) 17
Page 1 and 2: Boundary Conditions of the Hydrodyn
Page 3 and 4: fields are involved. This is the re
Page 5 and 6: ∂νΠ µν = 0 (7) and is symmetr
Page 7 and 8: Putting all this together, the Π 0
Page 9 and 10: (E, B) µν ⎛ ⎞ ⎜ 0 −Ex −
Page 11 and 12: 0= ˙ D + j D el + ρel v − c ∇
Page 13 and 14: The hydrodynamic equations are with
Page 15: vt = v − (v · n) n , (82) vn =(v
Page 19 and 20: ∆(D D n + Dn) =−σsf , (118)
Page 21 and 22: available, such as for a turbulent
Page 23 and 24: To obtain a finite βs, we add a th
Page 25 and 26: with H D z = αc A e−x/λ λ A =

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